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Full text of "Treatise On Analysis Vol-Ii"

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8 COMPLETE HILBERT ALGEBRAS 369 for all n, and all the AB have the same value y = (et \ e^) ~l. Hence, for all x9 y in A, we have (xan | yan) = (y*x \ an a*) = ( j;*x | yen) = y(xen \ yen) ; since the series with general term (xen \ yen) is absolutely convergent, with sum (x\y) (15.8.11), it follows that if A is infinite-dimensional then U{(x) is a Hilbert-Schmidt operator, and the relation (15.8.14.1) is valid. Since A is a Hilbert space, so is its image under U{ , and to show that this image is the whole of the Hilbert space JS?2(0 (15.4.8), it is enough to show that £7,(A) is dense in «^2(0- Now, for each pair, m, n with m ^ w, we have em A** ' *n A*i = *m(AO(A*i) = emAe1 (1 5.8.1 2(iii)), and since enAe1 = Can, it follows that there exists emn eemAen such that emnan = am (which implies that emn = y"1ama*)J and clearly ZmnVp = 0 if p^n. We conclude from this that £mij = t/r(O is the con- tinuous endomorphism of the Hilbert space I such that Emn • an = am and Emn ' ap = 0 if p ^ n. Our assertion now follows from the fact that the finite linear combinations of the Emn are dense in J$f2(0 (15.4.8). The proof is analogous but simpler when A is finite-dimensional. We remark that, in all cases, the C/r(In) consist of endomorphisms of the form U(x) oPn and therefore consist of endomorphisms of rank 1 of I. (15.8.15) Under the hypotheses of (15.8.1 4), if there exists an element ^0 in the center of A, then A Is finite-dimensional. In that case the center of A is Cw, where u Is the unit element of A. If c e A belongs to the center of A, then C/^c) is an endomorphism of the A-module I, hence is a homothety xi-+kx, where A e C (15.8.12). But clearly a homothety cannot be a Hilbert-Schmidt operator on an infinite-dimensional Hilbert space, unless it is zero. (15.8.16) Let A be a separable complete Hilbert algebra and let ak (keJ) be the topologically simple Hilbert algebras which are the Hilbert summands of A (1 5.8.1 3). For each k e J, let \k be a minimal left ideal ofak . Let Vbea non- degenerate representation (15.5.5) of A in a separable Hilbert space H, such that V : A i— > 2£ (H) is continuous. (i) H is the Hilbert sum ofsubspaces Hk (k e 3) stable with respect to F, such that if Vk is the restriction of V to Hk we have Vk(s) = Qfor all s e ah and all h 7^ k : so that Vk may be considered as a representation of ak on Hfc . (ii) If ak is finite-dimensional over C, the representation Vk is the Hilbert sum of a (finite or infinite) sequence of irreducible representations, each equiv- alent to the representation Uik ofak (15.8.14).he afc, that is to the case where A is